To take my mother as an example she hadn't thought about it properly at all. She was surprised that her daughter (my sister) had voted Remain because apparently my mother who knows about the EU funding which enabled what my sister did, hadn't connected "EU funding" to the EU she was voting to Leave. "Oh" was basically all she had to say about that.

In the broader scheme of things you could also see this in the response of Brexiteers who were stunned that, as non members they now weren't able to travel freely across borders.

Voters aren't very bright because they're people, you've met people. I used to think that FPTP was a good idea because it's so simple that "The candidate who had more votes won" means the idiots won't be confused by the outcome. We saw that they're confused anyway, so, fine, since apparently simplicity wasn't achievable we should use a more complicated voting system with better outcomes.

The idea of independence wasn't ridiculous, it's just the details made it hard to make it work.